A computational approach to linear conjugacy in a class of power law kinetic systems

Abstract

This paper studies linear conjugacy of PL-RDK systems, which are kinetic systems with power law rate functions whose kinetic orders are identical for branching reactions, i.e. reactions with the same reactant complex. Mass action kinetics (MAK) systems are the best known examples of such systems with reactant-determined kinetic orders (RDK). We specify their kinetics with their rate vector and T matrix. The T matrix is formed from the kinetic order matrix by replacing the reactions with their reactant complexes as row indices (thus compressing identical rows of branching reactions of a reactant complex to one) and taking the transpose of the resulting matrix. The T matrix is hence the kinetic analogue of the network's matrix of complexes Y with the latter's columns of non-reactant complexes truncated away. For MAK systems, the T matrix and the truncated Y matrix are identical. We show that, on non-branching networks, a necessary condition for linear conjugacy of MAK systems and, more generally, of PL-FSK (power law factor span surjective kinetics) systems, i.e. those whose T matrix columns are pairwise different, is T = T , i.e. equality of their T matrices. This motivated our inclusion of the condition T = T in exploring extension

A Deficiency-One Algorithm for power-law kinetic systems with reactant-determined interactions

“…Savageau [15] and Voit [18,19] highlighted the advantages of using power-law formalism for modelling biochemical systems. In this light, many CRN-based results on PLK systems are established ( [6,11,13,17] among others), some of which are extensions or modifications of existing results on MAK systems.…”

Linear conjugacy of chemical kinetic systems

“…Different networks could generate the same set of ODEs making them dynamically equivalent. In the past few years, various authors have pioneered the use of MILP algorithms for determining linear conjugacy between MAK systems (5 , 13 , 25 , 27 ), between rational functions systems (26 ), between GMAK systems (28 ) and between PL-RDK systems (7 ). In the work of (7 ), they extended the JSC for linear conjugacy from MAK systems to PL-RDK systems.…”

“…In relation to linear conjugacy, if the mapping h is trivial, M and M are said to be dynamically equivalent (7 ) .…”

“…The sparse linear conjugate of (N * , K * ) was obtained using the MILP algorithm, described in (7 ). The algorithm seeks to generate linearly conjugate realizations for a class of power-law kinetic systems, i.e., PL-RDK.…”

“…The three groups then collaborated in extending the MILP approach to linear conjugacy (a detailed discussion of the work up to 2013 can be found in (5 )). Further developments included the extension to "Bio-CRNs" (whose rate functions are mass action functions divided by positive polynomials in the species variables) by Gábor et al (6 ) and to complex factorizable power law kinetic systems (denoted by PL-RDK) by Cortez et al (7 ).…”